A065034 a(n) = Lucas(2*n) + 1.

3, 4, 8, 19, 48, 124, 323, 844, 2208, 5779, 15128, 39604, 103683, 271444, 710648, 1860499, 4870848, 12752044, 33385283, 87403804, 228826128, 599074579, 1568397608, 4106118244, 10749957123, 28143753124, 73681302248, 192900153619, 505019158608, 1322157322204

Offset: 0

Author

N. J. A. Sloane, Nov 04 2001

Keywords

• nonn
• easy

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
• Markus Grassl and Andrew J. Scott, Fibonacci-Lucas SIC-POVMs, arXiv:1707.02944 [quant-ph], 2017.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A000032, A000045, A005248, A005592.

Cf. A002878 (first differences). - R. J. Mathar, Jul 18 2009

Programs

Magma

[ Lucas(2*n) + 1: n in [0..210]]; // Vincenzo Librandi, Apr 15 2011

Maple

a:= n-> (<<0|1>, <1|1>>^(2*n). <<2,1>>)[1, 1]+1:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 01 2016

Mathematica

LucasL[2 Range[30]]+1 (* Harvey P. Dale, Oct 21 2011 *)
LinearRecurrence[{4, -4, 1}, {3, 4, 8}, 30] (* Jean-François Alcover, Jan 08 2019 *)

PARI

a(n) = { fibonacci(2*n + 1) + fibonacci(2*n - 1) + 1 } \\ Harry J. Smith, Oct 03 2009

PARI

Vec((3-2*x)*(1-2*x)/((1-x)*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Nov 01 2016

Formula

a(n) = F(2*n+1) + F(2*n-1) + 1 = A005248(n) + 1.

From R. J. Mathar, Jul 18 2009: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).

G.f.: 1/(1-x) + (2-3*x)/(1-3*x+x^2). (End)

a(n) = 1 + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n. - Colin Barker, Nov 01 2016

Extensions

a(0)=3 prepended by Joerg Arndt, Nov 01 2016